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Proof step 85 from proof prepost-extract-min (previous step)

ar0 = ar, bh = bin-heap(kar, m), bh0 = bh1.ar:= bh1.ar[0, bh1.ar[bh1.size]], bh1 = bin-heap(kar, m - 1), 2exp-size(# kar), bh1.size < # bh1.ar, 0 < # bh1.ar, 0 < # bh.ar, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar, m ≥ 1, ∃ key. key-index(key, bh.ar[0].idx) ∈ keyset(kar, m) ∧ (∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key) ⊦ # keyset(kar, m) = 0, bh1.size = 0, keyset(kar, m) = ∅, bh0.ar[0].idx < # ar ∧ 0 < # bh0.ar ∧ bh0.ar[0].idx < # ar ∧ 0 < # bh0.ar

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 86:
2exp-size(# kar), heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), key-index(key, kar[0].idx) ∈ keyset(kar, m), m ≤ # kar, # ar ≤ kar[m - 1].idx, ∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key ⊦ # keyset(kar, m) = 0, # kar = 0, keyset(kar, m) = ∅, m = 1, m = 0

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ # kar[n, ki] = # kar; (s)%m < # kar ⊦ kar[m, ki][m] = ki; (s)% ⊦ bin-heap(kar, m).ar:= kar0 = bin-heap(kar0, m); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ keyset(kar, 0) = ∅; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ 0 < m ↔ m ≠ 0; (s)% ⊦ m ≤ n → (n - m = 0 ↔ n = m); (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ bin-heap(kar, m).size = m; (s)